Level set learning with pseudo-reversible neural networks for nonlinear dimension reduction in function approximation

2021/12/02 by Yuankai Teng, Zhu Wang, Teng, Yuankai +7 · 2 citations
Computer Science · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2112.01438

openalex publication_date 2021/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), in this paper we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.

Citations

• Learning nonlinear level sets for dimensionality reduction in function approximation
• Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations

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